High-order finite-volume methods for the shallow-water equations on the sphere

Ullrich, P. A.; Jablonowski, Christiane; Leer, Bram van

doi:10.1016/j.jcp.2010.04.044

Cited by 90 publications

(117 citation statements)

References 52 publications

(83 reference statements)

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“…An angularly equidistant mapping [18] is used to generate the initial grids of the six adjoining grid sectors (or panels) that seamlessly cover each of the 2D spheres. In contrast to flows on 2D cubed-sphere grids, for which a curved coordinate system is normally defined on each of the six cubed-sphere sectors, the 3D cubed-sphere grid in principle allows the use of a unique coordinate system (e.g., Cartesian) to discretize the governing conservation laws everywhere in the physical domain, which makes unnecessary the usage of a covariant transformation [18,22,41,42] to map vector fields from the curved coordinate system to the Cartesian system.…”

Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

“…In recent years, cubed-sphere grids have gained increasing popularity for simulating fluid flow in domains between concentric spheres, first in the area of climate and weather modelling [18,19,20,21,22,23,24], but more recently also in areas like astrophysics [25,26]. Very recently, Ivan et al [14,15] have proposed a second-order parallel solution-adaptive computational framework for solving hyperbolic conservation laws on 3D cubed-sphere grids and applied the formulation to the simulation of several magnetized and nonmagnetized space-physics problems.…”

Section: Introductionmentioning

confidence: 99%

“…In spite of these challenges, high-order spatial discretizations on two-dimensional (2D) cubed-sphere grids have been successfully formulated for global atmospheric modelling by Ullrich et al [22] and Chen et al [24] (a) Cross-section of the cubed-sphere grid (b) Illustration of connectivity among root blocks Figure 1: Three-dimensional cubed-sphere grid with six root blocks (corresponding to the six sectors of the grid) and depiction of inter-block connectivity. In our approach, the root blocks can be refined in a block-adaptive way.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

High-order central ENO finite-volume scheme for ideal MHD

Susanto

Ivan

Sterck

et al. 2013

Journal of Computational Physics

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A high-order central essentially non-oscillatory (CENO) finite-volume scheme is developed for the compressible ideal magnetohydrodynamics (MHD) equations solved on threedimensional (3D) cubed-sphere grids. The proposed formulation is an extension to 3D geometries of a recent high-order MHD CENO scheme developed on two-dimensional (2D) grids. The main technical challenge in extending the 2D method to 3D cubed-sphere grids is to properly handle the nonplanar cell faces that arise in cubed-sphere grids. This difficulty is solved by considering general hexahedral cells with trilinear faces, which allow us to compute fluxes, areas and volumes with high-order accuracy by transforming to a reference cubic cell. The 3D CENO scheme is implemented within a flexible multi-block cubed-sphere grid framework to fourth-order accuracy, resulting in a high-order solution method for cubed-sphere grids with unique capabilities in terms of adaptive refinement and parallel scalability. The high-order method is applicable to the solution of general hyperbolic conservation laws with, in principle, arbitrary order. The CENO scheme is based on a hybrid solution reconstruction procedure that provides high-order accuracy in smooth regions, even for smooth extrema, and non-oscillatory transitions at discontinuities. The scheme is applied herein to MHD in combination with a GLM divergence correction technique to control the solenoidal condition for the magnetic field while preserving the high-order accuracy of the numerical procedure. The cubed-sphere simulation framework features a flexible design based on a genuine multi-block implementation, leading to high-order accuracy, flux calculation, adaptivity and parallelism that are fully transparent to the boundaries between the six sectors of the cubedsphere grid. The proposed 3D MHD CENO scheme is shown to achieve uniform fourth-order accuracy on cubed-sphere grids. Parallel domain partitioning and grid adaptivity are achieved on the 3D cubed-sphere grids using a hierarchical block-based division strategy with blocks of equal size. Numerical results to demonstrate the high-order accuracy, robustness and capability of the proposed high-order framework are presented and discussed for several test problems.

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Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-order central ENO finite-volume scheme for ideal MHD

Susanto

Ivan

Sterck

et al. 2013

Journal of Computational Physics

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“…Figure 10 illustrates the relative vorticity field of the barotropic wave at day 6 for all models. This could be compared with figure 19 of Ullrich et al (2010). The Yin-Yang grid (270 × 90 × 2) with t = 450 s does not capture the same wave solution as the others but forms the barotropic wave at resolution 405 × 135 × 2, equivalent to a resolution of 0.7 • at the Equator, with t = 900 s. This test is not easy at low resolution.…”

Section: Galewsky's Barotropic Wavementioning

confidence: 99%

Experiments with different discretizations for the shallow‐water equations on a sphere

Qaddouri

Pudykiewicz

Tanguay

et al. 2011

Quart J Royal Meteoro Soc

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“…A substantial amount of development was done in the computational aerodynamics community for external flows around complex bodies. The Cubed Sphere [31] is a type of multiblock grid that has also been developed for solving PDEs on a spherical surface; in [39], this grid is used with a high-order finite-volume method to solve the shallow-water equations. There is a rich literature on the subjects of mapped and multiblock grids that is too extensive to summarize here; we refer the interested reader to several review articles [35,37,2].…”

Section: Major Radiusmentioning

confidence: 99%

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids

McCorquodale

Dorr

Hittinger

et al. 2015

Journal of Computational Physics

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We present an approach to solving hyperbolic conservation laws by finitevolume methods on mapped multiblock grids, extending the approach of Colella, Dorr, Hittinger, and Martin (2011) for grids with a single mapping. We consider mapped multiblock domains for mappings that are conforming at inter-block boundaries. By using a smooth continuation of the mapping into ghost cells surrounding a block, we reduce the inter-block communication problem to finding an accurate, robust interpolation into these ghost cells from neighboring blocks. We demonstrate fourth-order accuracy for the advection equation for multiblock coordinate systems in two and three dimensions.

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High-order finite-volume methods for the shallow-water equations on the sphere

Cited by 90 publications

References 52 publications

High-order central ENO finite-volume scheme for ideal MHD

High-order central ENO finite-volume scheme for ideal MHD

Experiments with different discretizations for the shallow‐water equations on a sphere

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids

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